US2024346210A1PendingUtilityA1
Multi-scale analysis method for time series based on quantum walk
Est. expiryDec 9, 2041(~15.4 yrs left)· nominal 20-yr term from priority
G06F 2119/12G06F 30/27Y02A90/10G06N 10/00
35
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Claims
Abstract
Disclosed in the present disclosure is a multi-scale analysis method for time series based on quantum walk, including (1). generating multi-scale and multi-feature sequences based on quantum walk; (2). screening feature sequences; (3). modeling and predicting time series based on regression analysis; (4). performing frequency domain and time domain-based result evaluation; and (5). performing experimental verification. The method of the present disclosure has the advantage that the multi-scale features of the quantum walk are applied to the analysis of time series, and by combining feature extraction methods under two rules, a model of an original time series is established based on extracted features by using linear, nonlinear and time-based regression methods. The analysis method for time series does not require pre-assumption such as stationarity assumption, and is a universal analysis method for time series.
Claims
exact text as granted — not AI-modified1 . A multi-scale analysis method for time series based on quantum walk, specifically comprising following steps:
step 1. for an original observed time series, generating a plurality of feature sequences at different time scales based on quantum walk; step 2. performing feature selection on the plurality of feature sequences at different time scales generated in step 1 to obtain an optimal feature sequence combination; step 3. establishing a correlation model of the original observed time series and the optimal feature sequence combination based on a regression analysis method; and step 4. predicting an actual observed time series by using the correlation model in step 3, and prediction results are verified in the time domain and frequency domain.
2 . The multi-scale analysis method for time series based on quantum walk according to claim 1 , further comprising:
step 5. performing experimental verification on the multi-scale analysis method, wherein experimental configurations in the experimental verification are specifically as follows: experiment data configuration: satellites in a plurality of Pacific positions are selected, and absolute sea level data obtained by height measurement of the satellites is periodically collected, and then processed to obtain experimental data; and evaluation index configuration: a coefficient of determination R 2 , a root-mean-square error (RMSE) and a mean absolute error (MAE) are selected as evaluation indexes of the prediction result of the model, wherein the evaluation indexes are specifically expressed as follows:
R
2
=
1
-
∑
i
=
1
N
(
y
i
-
y
^
i
)
2
∑
i
=
1
N
(
y
i
-
y
_
)
2
RMSE
=
1
N
∑
i
=
1
N
(
y
^
i
-
y
i
)
2
MAE
=
1
N
∑
i
=
1
N
❘
"\[LeftBracketingBar]"
y
^
i
-
y
i
❘
"\[RightBracketingBar]"
wherein, y i is an i-th element of the actual observed time series, ŷ i is an i-th element of a sequence obtained by prediction fitting, y is an average value of elements of the actual observed time series, and N is the length of the time series.
3 . The multi-scale analysis method for time series based on quantum walk according to claim 1 , wherein step 1 is specifically implemented as follows:
representing a quantum walk process by an arbitrary undirected graph G=(V, E), wherein V is a set of vertices, and E is a set of edges; the vertices represent quantum states in the quantum walk process, and the edges represent transitions of the quantum states between the vertices; representing a quantum state vector at an initial moment in the quantum walk process by |φ(0) , and representing, utilizing a time evolution operator e −iHt , a quantum state vector |φ(t) at a moment t in the quantum walk process as:
❘
"\[LeftBracketingBar]"
φ
(
t
)
〉
=
e
-
ith
❘
"\[LeftBracketingBar]"
φ
(
0
)
〉
wherein, | is a symbol for labeling state vectors, e −iHt is the time evolution operator, i is an imaginary unit, and H is a Hamiltonian represented by an adjacency matrix or a Laplacian matrix;
decomposing a spectrum of the Hamiltonian H by using a spectrum decomposition algorithm to obtain eigenvalues and eigenvectors of the Hamiltonian H, wherein the decomposed Hamiltonian His:
H
=
ΦΛΦ
T
wherein, Φ is an N×N matrix, which represents a set of the eigenvectors; T represents a transposition; Λ is an N×N diagonal matrix, which is specifically expressed as Λ=diag (λ 1 , λ 2 , . . . , λ n , . . . , λ N ), λ 1 , λ 2 , . . . , λ N are the ordered eigenvalues of the Hamiltonian H; and N is the length of the time series;
the time evolution operator is expressed as e− iHt =Φe −iΛt Φ T , and then
the quantum state vector |φ(t) at the moment t in the quantum walk process is expressed as:
❘
"\[LeftBracketingBar]"
φ
(
t
)
〉
=
Φ
e
-
i
Λ
t
Φ
T
❘
"\[LeftBracketingBar]"
φ
(
0
)
〉
;
constructing a scale factor set {k j } j=1 J , wherein J represents the total number of scale factors, and k j represents a j-th scale factor; and when the moment t is replaced with k j n, the quantum state vector in the quantum walk process is expressed as:
❘
"\[LeftBracketingBar]"
φ
(
k
j
n
)
〉
=
Φ
e
-
j
Λ
k
j
n
Φ
T
❘
"\[LeftBracketingBar]"
φ
(
0
)
〉
,
k
j
∈
ℝ
+
wherein, + represents a positive real number, n is a natural number, n=0, 1, 2, . . . ; and
sampling the quantum walk process at an equal time interval based on the scale factor k j to obtain a sequence of norm squares of probability amplitudes corresponding to all the vertices, thereby generating the feature sequences of the quantum walk at different time scales.
4 . The multi-scale analysis method for time series based on quantum walk according to claim 3 , wherein the Hamiltonian H is represented by an adjacency matrix of graph G, and elements in the adjacency matrix of the graph G is expressed as:
A
uv
=
{
1
,
if
(
u
,
v
)
∈
E
0
,
otherwise
wherein, (u, v) represents an edge connecting a vertex u to a vertex v, A uv represents an edge between the vertex u and the vertex v, u∈V, v∈V, and A uv =A vu , and A vv =A uu =0.
5 . The multi-scale analysis method for time series based on quantum walk according to claim 1 , wherein in step 2, feature selection is performed on the generated plurality of feature sequences at different time scales by using stepwise regression, which is implemented as follows:
combining the feature sequences at different time scales, constantly adjusting the combinations, evaluating the fitting accuracy in using the combinations to model the original observed time series by using the Akaike information criterion, and selecting a combination with the best evaluation result as the optimal feature sequence combination; alternatively, feature selection is performed on the generated plurality of feature sequences at different time scales by using the RReliefF algorithm, which is implemented as follows: performing weight computation on the plurality of feature sequences at different time scales in step 1 based on the original observed time series, performing sorting according to the weights from large to small, and combining the first Q feature sequences at different time scales to form the optimal feature sequence combination.
6 . The multi-scale analysis method for time series based on quantum walk according to claim 1 , wherein the regression analysis method in step 3 comprises linear regression, nonlinear regression, or time-correlation-based vector autoregression methods, wherein the linear regression comprises but is not limited to stepwise regression, principal component regression, and partial least squares regression; and the nonlinear regression comprises but is not limited to projection pursuit regression.
7 . The multi-scale analysis method for time series based on quantum walk according to claim 6 , wherein in step 3, a correlation model of the original observed time series and the optimal feature sequence combination is established based on the linear regression, which is specified as follows:
Y
=
β
1
X
1
+
β
2
X
2
+
…
+
β
q
X
q
+
ε
wherein, Y is a fitted time series, X 1 , X 2 , . . . , X q are sequences in the optimal feature sequence combination respectively, β 1 , β 2 , . . . , β q are coefficients of the sequences respectively, and ε is a constant term.
8 . The multi-scale analysis method for time series based on quantum walk according to claim 6 , wherein in step 3, a correlation model of the original observed time series and the optimal feature sequence combination is established based on the projection pursuit regression, which is specified as follows:
F
(
x
)
~
∑
m
=
1
M
β
m
G
m
(
Z
m
)
=
∑
m
=
1
M
β
m
G
m
(
∑
p
=
1
P
a
mp
T
X
)
wherein, F(x) represents a fitted time series, G m (Z m ) represents a m-th ridge function, β m is a weight and represents the contribution of the m-th ridge function to an output value, M represents the total number of the ridge functions,
Z
m
=
∑
p
=
1
P
a
mp
T
X
is an independent variable of the m-th ridge function and represents a projection of a P-dimensional vector X in an α m direction, X represents high-dimensional data input in the model, α mp is a p-th component of the projection in the α m direction, a superscript T represents a transposition, P is a dimension of input space,
∑
p
=
1
P
a
p
2
=
1
is required, and α p represents a p-th component in a projection direction.
9 . The multi-scale analysis method for time series based on quantum walk according to claim 6 , wherein in step 3, a correlation model of the original observed time series and the optimal feature sequence combination is established based on the time-correlation-based vector autoregression, and the sequences in the optimal feature sequence combination are expressed in the form of a matrix as Y={X 1 , X 2 , . . . X w , . . . , X L }∈ N×L , w∈[1, L], which is implemented specifically as follows:
X
w
=
(
X
1
w
,
X
2
w
,
…
,
X
Nw
)
T
∈
ℝ
N
×
1
X
w
=
∑
z
=
1
d
A
z
X
w
-
z
+
ε
w
,
w
=
d
+
1
,
…
,
L
wherein, N represents the length of the time series, L represents the number of the sequences in the optimal feature sequence combination, X w represents vectors in a w-th column of a matrix Y, X w-z represents vectors in a w-z-th column of the matrix Y, X Nw represents an element value in the N-th row and the w-th column of the matrix Y, A z ∈ N×N is a coefficient matrix of the time-correlation-based vector autoregression, z is a lag order, d is a total lag order, and ε w represents noise.
10 . The multi-scale analysis method for time series based on quantum walk according to claim 1 , wherein in step 4, time-frequency domain-based result evaluation is performed on a prediction result, which is implemented specifically as follows:
selecting a coefficient of determination R 2 , a root-mean-square error (RMSE), and a mean absolute error (MAE) as evaluation indexes of the prediction result of the model, wherein the evaluation indexes are expressed as follows:
R
2
=
1
-
∑
i
=
1
N
(
y
i
-
y
^
i
)
2
∑
i
=
1
N
(
y
i
-
y
_
)
2
RMSE
=
1
N
∑
i
=
1
N
(
y
^
i
-
y
i
)
2
MAE
=
1
N
∑
i
=
1
N
❘
"\[LeftBracketingBar]"
y
^
i
-
y
i
❘
"\[RightBracketingBar]"
wherein, y i is an i-th element of the actual observed time series, ŷ i is an i-th element of a fitted sequence obtained by prediction, y is an average value of elements of the actual observed time series, and N is the length of the time series.Join the waitlist — get patent alerts
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